traveling and pinned fronts in bistable reaction ... · classical one-component reaction-diffusion...
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Traveling and Pinned Fronts in Bistable Reaction-Diffusion Systems on NetworksNikos E. Kouvaris1*, Hiroshi Kori2, Alexander S. Mikhailov1
1 Department of Physical Chemistry, Fritz Haber Institute of the Max Planck Society, Berlin, Germany, 2 Department of Information Sciences, Ochanomizu University,
Tokyo, Japan
Abstract
Traveling fronts and stationary localized patterns in bistable reaction-diffusion systems have been broadly studied forclassical continuous media and regular lattices. Analogs of such non-equilibrium patterns are also possible in networks.Here, we consider traveling and stationary patterns in bistable one-component systems on random Erdos-Renyi, scale-freeand hierarchical tree networks. As revealed through numerical simulations, traveling fronts exist in network-organizedsystems. They represent waves of transition from one stable state into another, spreading over the entire network. Thefronts can furthermore be pinned, thus forming stationary structures. While pinning of fronts has previously beenconsidered for chains of diffusively coupled bistable elements, the network architecture brings about significant differences.An important role is played by the degree (the number of connections) of a node. For regular trees with a fixed branchingfactor, the pinning conditions are analytically determined. For large Erdos-Renyi and scale-free networks, the mean-fieldtheory for stationary patterns is constructed.
Citation: Kouvaris NE, Kori H, Mikhailov AS (2012) Traveling and Pinned Fronts in Bistable Reaction-Diffusion Systems on Networks. PLoS ONE 7(9): e45029.doi:10.1371/journal.pone.0045029
Editor: Jurgen Kurths, Humboldt University, Germany
Received May 31, 2012; Accepted August 11, 2012; Published September 28, 2012
Copyright: � 2012 Kouvaris et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permitsunrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: Financial support from the German Research Foundation Collaborative Research Center SFB910 ‘‘Control of Self-Organizing Nonlinear Systems’’ (http://www.itp.tu-berlin.de/sfb910) and from the Volkswagen Foundation in Germany (http://www.volkswagenstiftung.de) is gratefully acknowledged. The funders hadno role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing Interests: The authors have declared that no competing interests exist.
* E-mail: [email protected]
Introduction
Studies of pattern formation in reaction-diffusion systems far
from equilibrium constitute a firmly established research field.
Starting from the pioneering work by Turing [1] and Prigogine
[2], self-organized structures in distributed active media with
activator-inhibitor dynamics have been extensively investigated
and various non-equilibrium patterns, such as rotating spirals,
traveling pulses, propagating fronts or stationary dissipative
structures could be observed [3,4]. Recently, the attention became
turned to network analogs of classical reaction-diffusion systems,
where the nodes are occupied by active elements and the links
represent diffusive connections between them. Such situations are
typical for epidemiology where spreading of diseases over
transportation networks takes place [5]. The networks can also
be formed by diffusively coupled chemical reactors [6] or
biological cells [7]. In distributed ecological systems, they consist
of individual habitats with dispersal connections between them [8].
Detailed investigations of synchronization phenomena in oscilla-
tory systems [9] and of infection spreading over networks [10]
have been performed. Turing patterns in activator-inhibitor
network systems have also been considered [11].
The analysis of bistable media is of principal importance in the
theory of pattern formation in reaction-diffusion systems. Travel-
ing fronts which represent waves of transition from one stable state
to another are providing a classical example of self-organized wave
patterns; they are also playing an important role in understanding
of more complex self-organization behavior in activator-inhibitor
systems and excitable media (see, e.g., [4,12]). The velocity and the
profile of a traveling front are uniquely determined by the
properties of the medium and do not depend on initial conditions.
Depending on the parameters of a medium, either spreading or
retreating fronts can generally be found. Stationary fronts, which
separate regions with two different stable states, are not
characteristic for continuous media; they are found only at special
parameter values where a transition from spreading to retreating
waves takes place. When discrete systems, formed by chains or
fractal structures of diffusively coupled bistable elements, are
considered, traveling fronts can however become pinned if
diffusion is weak enough, so that stable stationary fronts, which
are found within entire parameter regions, may arise [13–16].
In the present study, pattern formation in complex networks
formed by diffusively coupled bistable elements is numerically and
analytically investigated. Our numerical simulations, performed
for random Erdos-Renyi (ER) or scale-free networks and for
irregular trees, reveal a rich variety of time-dependent and
stationary patterns. The analogs of spreading and retreating fronts
are observed. Furthermore, stationary patterns, localized on
subsets of network nodes, are found. To understand such
phenomena, an approximate analytical theory for the networks
representing regular trees is developed. The theory yields the
bifurcation diagram which determines pinning conditions for trees
with different branching factors and for different diffusion
constants. Its results are used to interpret the behavior found in
irregular trees and for ER networks. Statistical properties of
stationary patterns in large random networks are moreover
analyzed in the framework of the mean-field approximation,
which has been originally proposed for spreading-infection
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problems [17–19] and has also been used in the analysis of Turing
patterns on the networks [11].
Bistable Systems on NetworksClassical one-component reaction-diffusion systems in continu-
ous media are described by equations of the form
_uu(x,t)~f (u)zD+2u(x,t) ð1Þ
where u(x,t) is the local activator density, function f (u) specifies
local bistable dynamics (see Methods) and D is the diffusion
coefficient. Depending on the particular context, the activator
variable u may represent concentration of a chemical or biological
species which amplifies (i.e. auto-catalyzes) its own production.
In the present study, we consider analogs of the phenomena
described by the model (1), which are however taking place on
networks. In network-organized systems, the activator species
occupies the nodes of a network and can be transported over
network links to other nodes. The connectivity structure of the
network can be described in terms of its adjacency matrix T whose
elements are Tij~1, if there is a link connecting the nodes i and j
(i,j~1,:::,N), and Tij~0 otherwise. We consider processes in
undirected networks, where the adjacency matrix T is symmetric
(Tij~Tji). Generally, the network analog of system (1) is given by
_uui~f (ui)zDXN
j~1
Tijuj{Tjiui
� �ð2Þ
where ui is the amount of activator in network node i and f (ui)describes the local bistable dynamics of the activator. The last term
in Eq. (2) takes into account diffusive coupling between the nodes.
Parameter D characterizes the rate of diffusive transport of the
activator over the network links.
Instead of the adjacency matrix, it is convenient to use the
Laplacian matrix L of the network, whose elements are defined as
Lij~Tij{kidij , where dij~1 for i~j, and dij~0 otherwise. In this
definition, ki is the degree, or the number of connections, of node i
given by ki~P
j Tji. In new notations Eq. (2) takes the form.
_uui~f (ui)zDXN
j~1
Lijuj : ð3Þ
When the considered network is a lattice, its Laplacian matrix
coincides with the finite-difference expression for the Laplacian
differential operator after discretization on this lattice.
A classical example of a one-component system exhibiting
bistable dynamics is the Schlogl model [20]. This model describes
a hypothetical trimolecular chemical reaction which exhibits
bistability (see Methods). In the Schlogl model, the nonlinear
function f (u) is a cubic polynomial
f (u)~{LV
Lu~{(u{r1)(u{r2)(u{r3) ð4Þ
so that V (u) has one maximum at r2 and two minima at r1 and r3.
We have performed numerical simulations and analytical inves-
tigations of the reaction-diffusion system (3) for different kinds of
networks using the Schlogl model.
Results
Numerical SimulationsIn this section we report the results of numerical simulations of
the bistable Schlogl model (3) for random ER networks and for
trees (the results for random scale-free networks are given in the
Supporting Information S1). The ER networks with the mean
degree SkT~7 and sizes N~150 or N~500 are considered. The
trees have several components with different branching factors.
The model (3) with the parameters r1~1 and r3~3 is chosen; the
parameter r2 and the diffusion constant D were varied in the
simulations. The parameter r2 was restricted to the interval
1vr2v2.
Traveling activation fronts were observed in ER networks. To
initiate such a front, a node at the periphery (with the minimum
degree k) could be chosen and set into the active state u~r3,
whereas all other nodes were in the passive state u~r1. This
configuration was found to generate a wave of transition from the
passive to the active states. The wave spreads from the initially
active node to the rest of the system and reaches equidistant nodes,
located at the same distance (the shortest path length) from the
initial node, at about the same time.
Front propagation is illustrated in Fig. 1, where the nodes are
grouped according to their distance from the first activated node
and the average value rh of the activator density u in each group is
plotted as a function of the distance h. Three snapshots of the
traveling front at different times are displayed. As we see, for
increasing time the front moves into the subsets of nodes with the
larger distances. At the end, all nodes are in the active state r3.
Not all initial conditions lead, however, to spreading fronts. If
for example, for the same model parameters as in Fig. 1, a hub
node was initially activated, a spreading activation front could not
be produced. Retreating fronts were found at these parameter
values if the initial activation was set in a few neighbor nodes with
large degrees. Under weak diffusive coupling, stationary localized
patterns were furthermore observed. If the initial activation was set
on the nodes with moderate degrees, the activation could neither
spread nor retreat, thus staying as a stationary localized structure.
On the other hand, traveling fronts could also become pinned
when some nodes were reached, so that the activation could not
spread over the entire network and stationary patterns with
coexistence of the two states were established.
Degrees of the nodes play an important role in front pinning. In
the representation used in Fig. 2, the nodes with higher degrees lie
in the center, whereas the nodes with small degrees are located in
the periphery of the network. To produce the stationary pattern
shown in this figure, some of the central hub nodes were set into
the active state r3, while all other nodes were in the passive state r1.
The activation front started to propagate towards the periphery,
but the front became pinned and a stationary pattern was formed.
Figure 3 shows another example of a stationary pattern. Here, we
have sorted network nodes according to their degrees, so that the
degree of a node becomes higher as its index i is increased (the
stepwise red curve indicates the degrees of the nodes). Localization
on a subset of the nodes with high degrees is observed.
The importance of the degrees of the nodes becomes
particularly clear when front propagation in the trees with various
branching factors is considered (in a tree, the branching factor of a
node with degree k is k{1). The networks shown in Fig. 4 consist
of the component trees with the branching factors 2,3,4 and 5which are connected at their origins. If the activation is initially
applied to the central node, it spreads for D~0:1 through the trees
with branching factors 2 and 3, but cannot propagate through the
trees with higher branching factors (Fig. 4A and Movie S1). If we
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Figure 1. Traveling front in an Erdos-Renyi network. The network size is N~500 and the mean degree is SkT~7. Three consequent snapshotsof activity patterns at times t~0,10,21 are shown. Quantity rh is the average value of the activator density u in the subset of network nodes locatedat distance h from the node which was initially activated. Other parameters are r1~1, r2~1:2, r2~3; the diffusion constant is D~0:1.doi:10.1371/journal.pone.0045029.g001
Figure 2. Stationary pattern in an Erdos-Renyi network. The network size is N~150 and the mean degree is SkT~7. The nodes with higherdegrees are located closer to the center. The nodes are colored according to their activation level, as indicated in the bar. The other parameters arer1~1, r2~1:4 and r3~3; the diffusion constant is D~0:01.doi:10.1371/journal.pone.0045029.g002
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choose a larger diffusion constant D~0:35 and apply activation to
a subset of nodes inside the tree with the branching factor 5, the
activation retreats and dies out (Fig. 4B). When diffusion is weak
(D~0:03), the application of activation inside the component trees
leads to its spreading towards the roots of the trees. The activation
cannot however propagate further and pinned stationary struc-
tures are formed (Fig. 4C and Movie S2).
Thus, we see that both traveling fronts and pinned stationary
structures can be observed in the networks. Our numerical
simulations suggest that degrees of the nodes (and the related
branching factors in the trees) should play an important role in
such phenomena. The observed behavior is however complex and
seems to depend on the architecture of the networks and on how
the initial activation was applied. Below, it is analytically
investigated for regular trees with fixed branching factors. The
approximate mean-field description for stationary patterns in large
random networks is moreover constructed. Using analytical
results, complex behavior observed in numerical simulations can
be understood.
Front Dynamics in Regular TreesLet us consider the model (3) for a regular tree with the
branching factor k{1. In such a tree, all nodes, lying at the same
distance l from the origin, can be grouped into a single shell and
front propagation along the sequence of the shells l~1,2,3,::: can
be studied. Suppose that we have taken a node which belongs to
the shell l. This node should be diffusively coupled to k{1 nodes
in the next shell lz1 and to just one node in the previous shell
l{1. Introducing the activation level ul in the shell l, the evolution
of the activator distribution on the tree can therefore be described
by the equation
_uul~f (ul)zD(ul{1{ul)zD(k{1)(ulz1{ul) : ð5Þ
Note that for k~2, Eq. (5) describes front propagation in a one-
dimensional chain of coupled bistable elements. Propagation
failure and pinning of fronts in chains of bistable elements have
been previously investigated [13–15]. The approximate analytical
theory for front pinning in the trees, which is presented below,
represents an extension of the respective theory for the chains [15].
Note furthermore that model (5) can be formally considered for
any values of kw2 of the parameter k (but actual trees correspond
only to the integer values of this parameter).
Comparing the situations for the chains of coupled single elements
and for coupled shells in a tree (Eq. (5)), an important difference
should be stressed. In a chain, both propagation directions (left or
right) are equivalent, because of the chain symmetry. In contrast to
this, an activation front propagating from the root to the periphery of
a tree is physically different from the front propagating in the
opposite direction, i.e. towards the tree root. As we shall soon see,
one of such fronts can be spreading while the other can be pinned or
retreating for the same set of model parameters.
The approximate analytical theory of front pinning can be
constructed (cf. [15]) if diffusion is weak and the fronts are very
narrow. A pinned front is found by setting _uul~0 in Eq. (5), so that
we get
f (ul)zD(ul{1{ul)zD(k{1)(ulz1{ul)~0 : ð6Þ
Suppose that the pinned front is located at the shell l~m and it
is so narrow that the nodes in the lower shells lvm are all
approximately in the active state r3, whereas the nodes in the
higher shells are in the passive state r1. Then, the activation level
um in the interface l~m should approximately satisfy the
condition
g(um)~f (um)zD (k{1)r1{kumzr3½ �~0 : ð7Þ
Thus, the problem becomes reduced to finding the solutions of
Eq. (7). When D~0, we have g(um)~f (um) and, therefore, Eq. (7)
has three roots um~r1,r2,r3; the front is pinned then. Equation (7)
has also three roots if D is small enough (see Fig. 5 for D~0:03). In
this situation, the front continues to be pinned. Under further
increase of the diffusion constant (see Fig. 5 for D~0:1), the two
smaller roots merge and disappear, so that only one (larger) root
remains. As previously shown for one-dimensional chains of
diffusively coupled elements [15], such transition corresponds to
the disappearance of pinned stationary fronts.
The transition from pinned to traveling fronts takes place
through a saddle-node bifurcation. When k is fixed, the
bifurcation occurs when some critical value of D is exceeded (see
Fig. 6A). If the diffusion constant is fixed, pinned fronts are found
inside an interval of degrees k (see Fig. 6B).
Generally, the bifurcation boundary can be determined from
the conditions g(um)~0 and g0(um)~0, which can be written in
the parametric form as
D~2um{r2{r3ð Þ r1{umð Þ2
r1{r3
k~{3u2
mz2 r1zr2zr3ð Þum{r1r2{r1r3{r2r3
D: ð8Þ
Equations (8) determine boundaries between regions II and III
or II and IV in the bifurcation diagram in Fig. 7. The two
boundaries merge in the cusp point, which is defined by the
conditions g(um)~g0(um)~g
00(um)~0 and is located at
Figure 3. Nodes activation levels for a stationary pattern in anErdos-Renyi network. Dependence of the activation level ui on thedegrees ki of the nodes i is presented for a stationary pattern in the ERnetwork of size N~500 and mean degree SkT~7. The nodes areordered according to their increasing degrees, shown by the stepwisered curve. The other parameters are r1~1, r2~1:4 and r3~3; thediffusion constant is D~0:01.doi:10.1371/journal.pone.0045029.g003
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Dcusp~r3zr2{2r1ð Þ3
27 r3{r1ð Þkcusp~
r21zr2
2zr23{r1r2{r1r3{r2r3
3Dcuspð9Þ
in the parameter plane.
Figure 4. Spreading, retreating and pinning of activation fronts in trees. A) For D~0:1, the fronts spread to the periphery through the nodeswith the degrees k~2,3,4, while they are pinned at the nodes with the larger degrees. B) For D~0:35, the front is retreated from nodes with degreek~6. C) For D~0:03, the fronts propagate towards the root, but not towards the periphery. In each row, the initial configuration (left) and the finalstationary pattern (right) are displayed. The same color coding for node activity as in Fig. 2 is used. Other parameters are r1~1, r2~1:4 and r3~3.doi:10.1371/journal.pone.0045029.g004
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Above the cusp point, there should be a boundary line
separating regions III and IV. Indeed, fronts propagate in opposite
directions in these two regions and, to go from one to another, one
needs to cross a line on which the propagation velocity vanishes.
This boundary can be identified by using the arguments given
below.
Suppose that the diffusion constant is fixed and DvDcusp. Then
the pinned fronts are found inside an interval of degrees k, where
equation g(um)~0 has three roots, as in Fig. 8A for k~7. Outside
this interval, equation g(um)~0 has a single root, which
corresponds to spreading fronts if it is close to r3 (Fig. 8A for
k~4) or to retreating fronts if it is close to r1 (Fig. 8A for k~9).
Thus, if we traverse the bifurcation diagram in Fig. 7 below Dcusp
by increasing k, function g(um) will change its form as shown in
Fig. 8, having three zeroes within an entire interval of degrees k
that corresponds to the pinning region II. When the diffusion
constant is increased and the cusp at D~Dcusp is approached,
such interval shrinks to a point. If we traverse the bifurcation
diagram in Fig. 7 above the cusp, the function g(um) changes as
shown in Fig. 8B. For a given diffusion constant D, there is only
one degree k, such that the function g(um) has an inflection point
coinciding with its zero. The boundary separating regions III and
IV is determined by the conditions g(um)~0 and g00(um)~0. In
the parameter plane, these conditions yield the curve
D~r1zr2{2r3ð Þ r2zr3{2r1ð Þ r1zr3{2r2ð Þ
9 3(r3{r1){ r3zr2{2r1ð Þk½ � ð10Þ
where the propagation velocity of the fronts is changing its sign.
The above results refer to the first kind of fronts, where the
nodes in the lower shells (lvm) of the tree are in the active state
u~r3 and the nodes in the periphery are in the passive state u~r1.
A similar analysis can furthermore be performed for the second
kind of fronts, where the nodes in the periphery are in the active
state and the nodes in the lower shells are in the passive state. Such
pinned fronts are again determined by equation (7), where
however the parameters r3 and r1 should be exchanged. The
pinning boundary for them can be obtained from equations (8)
under the exchange of r1 and r3. This yields.
D~r1zr2{2umð Þ r3{umð Þ2
r1{r3
Figure 5. Functions g(um) for three different values of D. Theother parameters are r1~1, r2~1:4,r3~3 and k~4.doi:10.1371/journal.pone.0045029.g005
Figure 6. The roots of Eq. (7). The roots um are plotted as functions (A) of the diffusion constant D for k~4 and (B) of the degree k for D~0:1.Pinned fronts correspond to red parts of the curves. The model parameters are r1~1, r2~1:4 and r3~3.doi:10.1371/journal.pone.0045029.g006
Figure 7. The bifurcation diagram. Regions with differentdynamical regimes are shown in the parametric plane k{D. Blackcurves indicate the saddle-node bifurcations given by the Eq. (8), whilethe blue curve stands for the saddle-node bifurcations given by Eq. (11).The green dot indicates the cusp point given in Eq. (9), the red curveshows the boundary determined by Eq. (10). The model parameters arer1~1, r2~1:4 and r3~3.doi:10.1371/journal.pone.0045029.g007
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k~{3u2
mz2 r1zr2zr3ð Þum{r1r2{r1r3{r2r3
D: ð11Þ
Fronts of the second kind are pinned for sufficiently weak
diffusion, inside region I in the bifurcation diagram in Fig. 7. The
boundary of this region is determined in the parametric form by
Eqs. (11).
Thus, our approximate analysis has allowed us to identify
regions in the parameter plane (k,D) where the fronts of different
kinds are pinned or propagate in specific directions. Predictions of
the approximate analytical theory agree well with numerical
simulations for regular trees. Figure 9 shows traveling and pinned
fronts found by direct integration of Eq. (5) in different regions of
the parameter plane. For each region, the behavior of two kinds of
the fronts, with the activation applied to the nodes of the lower
shells (lƒ6) or periphery nodes (lw6), is illustrated. When the
parameters D and k are chosen within region I of the bifurcation
diagram, both kinds of fronts are pinned (Figs. 9A(I),B(I)). In
region II, the front initiated from the tree origin is pinned
(Fig. 9A(II)), whereas the front initiated in the periphery
propagates towards the root (Fig. 9B(II)). Activation fronts which
propagate in both directions, towards the root and the periphery,
are found in region III (Fig. 9A(III),B(III)). In region IV, the
activation front initiated at the root is retreating (Fig. 9A(IV)),
whereas the front initiated at the periphery is spreading
(Fig. 9B(IV)).
In addition to providing examples of the front behavior, Fig. 9
also allows us to estimate the accuracy of approximations used in
the derivation of the bifurcation diagram. In this derivation, we
have assumed (similar to Ref. [15]) that diffusion is weak and the
fronts are so narrow that only in a single point the activation level
differs from its values r1 and r3 in the two uniform stable states.
Examining Fig. 9, we can notice that this assumption holds well for
the lowest diffusion constant D~0:01 in region I, whereas
deviations can be already observed for the faster diffusion in
regions II, III and IV. Still, the deviations are relatively small and
the approximately analytical theory remains applicable.
Figure 10 shows the numerically determined propagation
velocity of both kinds of activation fronts for different degrees kat the same diffusion constant D~0:1. The blue curve
corresponds to the fronts of the first kind, with the activation
applied at the tree root. Such front is spreading towards the
periphery for small degrees k (region III), is pinned in an interval
of the degrees corresponding to region II, and retreats towards the
root for the larger values of k (region IV). The red curve displays
the propagation velocity for the second kind of fronts, with the
activation applied at the periphery. For the chosen value of the
diffusion constant, such front is always spreading, i.e. moving
towards the root. We can notice that, for the same parameter
values, the absolute propagation velocity of the second kind of
fronts is always higher than that for the fronts of the first kind.
Using Fig. 9, we can consider evolution of various localized
perturbations in different parts of the bifurcation diagram (Fig. 7).
Inside region I, all fronts are pinned. Therefore, any localized
perturbation (see Fig. 11A(I),B(I)) is frozen in this region. If the
activation is locally applied inside region II, it spreads towards the
root, but cannot spread in the direction to the periphery
(Fig. 11A(II)). On the other hand, if a local ‘‘cold’’ region is
created in region II on the background of the ‘‘hot’’ active state, it
shrinks and disappears (Fig. 11B(II)). In region III, local activation
spreads in both directions, eventually transferring the entire tree
into the hot state (Fig. 11A(III)), whereas the local cold region on
the hot background shrinks and disappears (Fig. 11 B(III)). An
interesting behavior is found in region IV. Here, both kinds of
fronts are traveling in the same direction (towards the root), but
the velocity of the second of them is higher (cf. Fig. 10). Therefore,
the hot domain would gradually broaden while traveling in the
root direction (Fig. 11A(IV)). The local cold domain (Fig. 11B(IV))
would be however shrinking while traveling in the same direction.
With these results, complex behavior observed in numerical
simulations for the trees with varying branching factors (Fig. 4) can
be understood. In the simulation shown in Fig. 4A, the diffusion
constant was D~0:1 and, according to the bifurcation diagram in
Fig. 7, the nodes with degrees k~2,3,4 should correspond to
region III, while the nodes with the higher degrees k~5,6 are in
the region II. Indeed, we can see in Fig. 4A that activation can
propagate from the root over the subtrees with the small
branching factors, but the front fails to propagate through the
subtrees with node degrees 5 and 6. In the simulation for D~0:35shown in Fig. 4B, the activation has been initially applied to a
group of nodes with degree k~6 corresponding to region IV. In
accordance with the behavior illustrated in Fig. 11A(IV), such local
perturbations broaden while traveling towards the root of the tree,
but get pinned and finally disappear. In Fig. 4C we have D~0:03and, therefore, we are in region II for all degrees k. According to
Fig. 11A(II), local activation in any component tree should spread
Figure 8. The typical form of functions g(um) in different regions of the parameter plane. Functions g(um) are shown below(A, D~0:1) and above (B, D~0:32) the cusp point. The green curve in part (B) corresponds to the boundary between regions III and IV, where thefront propagation velocity vanishes. The other parameters are r1~1, r2~1:4 and r3~3.doi:10.1371/journal.pone.0045029.g008
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towards the root, but cannot propagate towards the periphery, in
agreement with the behavior illustrated in Fig. 4C.
Although our study has been performed for the trees, its results
can also be used in the analysis of front propagation in large
random Erdos-Renyi networks. Indeed, it is known [21] that the
ER networks are locally approximated by the trees. Hence, if the
initial perturbation has been applied to a node and starts to spread
over the network, its propagation is effectively taking place on a
tree formed by the node neighbors. Previously, we have used this
property in the analysis of oscillators entrainment by a pacemaker
in large ER networks [22,23]. Only when the activation has
already covered a sufficiently high fraction of the network nodes,
loops start to play a role. When this occurs, the activation may
arrive at a node along different pathways and the tree
approximation ceases to hold. In this opposite situation, a different
theory employing the mean-field approximation can however be
applied.
Mean-field ApproximationRandom ER networks typically have short diameters and
diffusive mixing in such networks should be fast. Under the
conditions of ideal mixing, the mean-field approximation is
applicable; it has previously been used to analyze epidemic
spreading [17–19], limit-cycle oscillations and turbulence [24], or
Turing patterns [11] on large random networks. In this
approximation, details of interactions between neighbors are
neglected and each individual node is viewed as being coupled to a
global mean field which is determined by the entire system. The
network nodes contribute to the mean field according to their
degrees k. The strength of coupling of a node to such global field
and also the amount of its contribution to the field are not the
same for all nodes and are proportional to their degrees. Thus, a
node with a higher number of connections is more strongly
affected by the mean field, generated by the rest of the network,
and it also contributes stronger to such field. The mean-field
approximation is applied below to analyze statistical properties of
stationary activation patterns which are well spread over a network
and involve a relatively large fraction of nodes.
Similar to publications [11,24], we start by introducing the local
field
Figure 9. Stationary and traveling fronts in regular trees. The arrows show the propagation direction for traveling fronts. The labels refer todifferent regions of the bifurcation diagram in Fig. 7. They correspond to the parameter values (I) k~3, D~0:01, (II) k~6, D~0:03, (III) k~3, D~0:1and (IV) k~12, D~0:1. The other parameters are r1~1, r2~1:4 and r3~3.doi:10.1371/journal.pone.0045029.g009
Figure 10. Dependence of the front velocity on node degree.The blue curve corresponds to the first kind of fronts, shown in Fig. 9A;the red curve is for the second kind of fronts shown in Fig. 9B. Thediffusion constant is D~0:1; the other parameters are r1~1, r2~1:4and r3~3.doi:10.1371/journal.pone.0045029.g010
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qi~XN
j~1
Tijuj ð12Þ
determined by the activation of the first neighbors of a network
node i. Then, the evolution equation (2) can be written in the form
_uui~f (ui)zD(qi{kiui) ð13Þ
so that it describes the interaction of the element at node i with the
local field qi.
The mean-field approximation consists of the replacement of
the local fields qi by qi~kiQ, where the global mean field is
defined as
Q~XN
j~1
wjuj : ð14Þ
Here, the weights
wj~kjPN
n~1 kn
ð15Þ
guarantee that the nodes with higher degrees ki contribute
stronger to the mean field. After such replacement, Eq. (13) yields
_uu~f (u)zb(Q{u) ð16Þ
where b~Dk. Note that the index i could be removed because the
same equation holds for all network nodes.
Equation (16) describes bistable dynamics of an element coupled
to the mean field Q. The coupling strength is determined by the
parameter b which is proportional to the degree k of the
considered node. According to Eq. (16), behavior of the elements
located in the nodes with small degrees (and hence small b) is
mostly determined by local bistable dynamics, whereas behavior of
the elements located in the nodes with large degrees (and large b) is
dominated by the mean field.
The fixed points of Eq. (16) yield activator levels u in single
nodes coupled with strength b to the mean field Q. Self-organized
stationary patterns on a random ER network can be analyzed in
terms of this mean-field equation. Indeed, the activator level in
each node i of a pattern can be calculated from Eq. (16), assuming
that the node is coupled to the mean field determined by the entire
network. In Fig. 12, the mean-field approximation is applied to
analyze the stationary pattern shown in Fig. 3. This pattern has
developed in the ER network of size N~500 and mean degree
SkT~7 when the diffusion constant was fixed at D~0:01. The
mean field corresponding to such pattern was computed in direct
numerical simulations and is equal to Q~1:5. Substituting this
value of Q into Eq. (16), activator levels u in single node, coupled
to this mean field can be obtained. In Fig. 12A, the activator level
u is plotted as a function of the parameter b. When a node is
decoupled (b~0), Eq. (16) has three fixed points r1,r2,r3. As b is
increased, the system undergoes a saddle-node bifurcation beyond
which only one stable fixed point remains.
According to the definition of the parameter b, each node i with
degree ki is characterized by its own value bi~Dki of this
bifurcation parameter. Therefore, the fixed points of Eq. (16) can
be used to determine the activation levels for each node i, if its
Figure 11. Nonlinear evolution of local perturbations. The evolution of different local perturbations (A,B) is shown in various regions of thebifurcation diagram. The arrows show the propagation direction. The parameter values are (I) k~3, D~0:01, (II) k~6, D~0:03, (III) k~3, D~0:1and (IV) k~12, D~0:1. The other parameters are r1~1, r2~1:4 and r3~3.doi:10.1371/journal.pone.0045029.g011
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degree ki is known. The stationary activity distributions, predicted
by the mean-field theory and found in direct numerical
simulations, are displayed in Fig. 12B, where the nodes are
ordered according to their increasing degrees. Note that the value
Q~1:5 of the mean field, used to determine the activity levels, has
been taken here from the numerical simulation. As we see, data
points indeed lie on the two stable branches of the bifurcation
diagram, indicating a good agreement with the mean-field
approximation. In the Supporting Information S1, a similar
mean-field analysis is performed for self-organized stationary
activity patterns on scale-free networks.
Discussion
Traveling fronts represent classical examples of non-equilibrium
patterns in bistable reaction-diffusion media. As shown in our
study, such patterns are also possible in networks of diffusively
coupled bistable elements, but their properties are significantly
different. In addition to spreading or retreating activation fronts,
stationary fronts are found within large parameter regions. The
behavior of the fronts is highly sensitive to network architecture
and degrees of network nodes play an important role here.
In the special case of regular trees, an approximate analytical
theory could be constructed. The theory reveals that branching
factors of the trees and, thus, the degrees of their nodes, are
essential for front propagation phenomena. By using this
approach, front pinning conditions could be derived and
parameter boundaries, which separate pinned and traveling fronts,
could be determined. As we have found, propagation conditions
are different for the fronts traveling from the tree root to the
periphery or in the opposite direction. Generally, all fronts become
pinned as the diffusion constant is gradually reduced. While the
theory has been developed for regular trees, where the branching
factor is fixed, it is also applicable to irregular trees where node
degrees are variable. Indeed, at sufficiently weak diffusion the front
pinning occurs locally and its conditions are effectively determined
only by the degrees of the nodes at which a front becomes pinned.
The results of such analysis are relevant for understanding the
phenomena of activation spreading and pinning in large random
networks. It is well known (see, e.g., [21]) that, in the large size
limit, random networks are locally approximated by the trees. If
the number of connections (the degree) of a node is much smaller
than the total number of nodes in a network, the probability that a
neighbor of a given node is also connected to another neighbor of
the same node is small, implying that the local pattern of
connections in the vicinity of a node has a tree structure. This
property holds as long as the number of nodes in the considered
neighborhood is still much smaller that the total number of nodes
in the network. Previously, the local tree approximation has been
successfully used in the analysis of pacemakers in large random
oscillatory networks [22,23].
When activation is applied to a node in a large random network,
it spreads through a subnetwork of its neighbors and, at sufficiently
short distances from the original node, such subnetwork should be
a tree. Hence, our study of front propagation on the trees is also
providing a theory for the initial stage of front spreading from a
single activated node in large random networks. Depending on the
diffusion constant and other parameters, the fronts may become
pinned while the activation has not yet spread far away from the
original node. Whenever this takes place, the approximate pinning
theory, constructed for the trees, is applicable.
On the other hand, if the activation spreads far from the origin
and a large fraction of network nodes become thus affected, the
patterns can be well understood with the mean-field approxima-
tion. This approximation, proposed in the analysis of infection
spreading on networks [17], has also been applied to analyze
Turing patterns in network-organized activator-inhibitor systems
[11] and effects of turbulence in oscillator networks [24]. In this
paper, we have applied this approximation to the analysis of
stationary activity distributions in random Erdos-Renyi and scale-
free networks of diffusively coupled bistable elements. We could
observe that, within the mean-field approximation, statistical
properties of network activity distributions are well reproduced. It
should be noted that, similar to previous studies [11,24], the mean-
field values used in the theory were taken from direct numerical
simulations and were not obtained through the solution of a
consistency equation. Hence, we could only demonstrate that such
an approximation is applicable for the statistical description of the
emerging stationary patterns, but did not use it here for the
prediction of such patterns.
Figure 12. The stationary activity pattern and the mean-field bifurcation diagram. (A) The bifurcation diagram of Eq. (16) for the meanfield Q~1:5. (B) Activity distribution in the stationary pattern in the ER network of size N~500 and mean degree SkT~7 at D~0:01 is comparedwith the activator levels u predicted by the mean-field theory for Q~1:5. Blue crosses show the simulation data. Black and red curves indicate stableand unstable fixed points of the mean-field equation (16). The other parameters are r1~1, r2~1:4, r3~3.doi:10.1371/journal.pone.0045029.g012
Traveling and Pinned Fronts in Network Systems
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Thus, our investigations have shown that a rich behavior
involving traveling and pinned fronts is characteristic for networks
of diffusively coupled bistable elements. In the past, pinned fronts
were observed in the experiments using weakly coupled bistable
chemical reactors on a ring [25,26]. It will be interesting to
perform similar experiments for the networks of coupled chemical
reactors. Recent developments in nanotechnology allow to design
chemical reactors at the nanoscale and couple them by diffusive
connections to build networks [6]. It should be also noted that,
while the chemical Schlogl model has been used in our numerical
simulations, the results are general and applicable to any networks
formed by diffusively coupled bistable elements. The phenomena
of front spreading and pinning should be possible for diffusively
coupled ecological populations and similar effects may be involved
when epidemics spreading under bistability conditions is consid-
ered.
Methods
Bistable DynamicsThe Schlogl model [20] corresponds to a hypothetical reaction
scheme
Az2X /?c2
c13X
X /?c4
c3B : ð17Þ
If concentrations of reagents A and B are kept fixed, the rate
equation for the concentration u of the activator species X reads.
_uu(t)~{c2u3(t)zc1au2(t){c3u(t)zc4b ð18Þ
where the coefficients c1,c2,c3,c4 are rate constants of the
reactions; a~½A�, b~½B� and u~½X � are concentrations of
chemical species. By choosing appropriate time units, we can set
c2~1. Then, the right side of Eq. (18), can be written as
f (u)~{(u{r1)(u{r2)(u{r3) ð19Þ
where the parameters r1,r2,r3 satisfy the conditions
c1~r1zr2zr3
a
c3~r1r2zr1r3zr2r3
c4~r1r2r3
b: ð20Þ
The cubic polynomial f (u) has three real roots which
correspond to the steady states (fixed points) of the dynamical
system (18).
NetworksErdos-Renyi networks were constructed by taking a large number
N of nodes and randomly connecting any two nodes with some
probability p. This construction algorithm yields a Poisson degree
distribution with the mean degree SkT~pN [27]. In our study we
have considered the largest connected component network, namely,
we have removed the nodes with the degree k~0.
Tree networks with branching factor k{1 were constructed by
a simple iterative method. We start with a single root node and at
each step add k{1 nodes to each existing node with the degree
k~1. After L steps this algorithm leads to a tree network with the
size N~PL
l~1 (k{1)l{1, where the root node has degree k{1,
the last added nodes have degree 1 and all other nodes have
degree k. In our numerical simulations we have also used complex
trees consisting of component trees with different fixed branching
factors which are connected at their origins.
Scale-free networks, considered in the Supporting Information
S1, were constructed by the preferential attachment algorithm of
Barabasi and Albert [27]. Starting with a small number of m nodes
with m connections, at each next time step a new node is added,
with m links to m different previous nodes. The new node will be
connected to a previous node i, which has ki connections, with the
probability ki=P
j kj . After many time steps, this algorithm leads
to a network composed by N nodes with the power-law degree
distribution P(k)*k{3 and the mean degree vkw~2m.
To display the networks in Figs. 2, 4 and S2B we have used the
Fruchterman-Reingold force-directed algorithm which is available
in the open-source Python package NetworkX [28]. This network
visualization algorithm places the nodes with close degrees k near
one to another in the network projection onto a plane.
Numerical MethodsFor networks of coupled bistable elements, simulations were
carried out by numerical integration of Eq. (3) using the explicit
Euler scheme
u(tz1)i ~u
(t)i zdt f u
(t)i
� �zD
XN
j~1
Liju(t)j
" #ð21Þ
with the time step dt~10{3. The integration was performed for
5|105 steps. The initial conditions were ui~1 for all network
nodes i, except a subset of nodes to which initial activation was
applied and where we had ui~3. The explicit Euler scheme with
the same time step dt was also used to integrate Eq. (5) which
describes patterns on regular trees.
Supporting Information
Supporting Information S1 The results of the numericalsimulations of the bistable Schlogl model (3) for scale-free networks are provided. Traveling fronts and stationary
localized patterns are reported for networks with mean degree
SkT~6 and sizes N~150 or N~500 nodes. The observed
stationary pattern is compared with the mean-field bifurcation
diagram.
(PDF)
Movie S1 This movie shows time evolution startingfrom the initial conditions in Fig. 4A (left). For numerical
simulations, we have used the parameters D~0:1, r1~1, r2~1:4and r3~3. The same color coding for the node activity as in Fig. 2
is applied.
(MP4)
Traveling and Pinned Fronts in Network Systems
PLOS ONE | www.plosone.org 11 September 2012 | Volume 7 | Issue 9 | e45029
Movie S2 This movie shows time evolution startingfrom the initial conditions in Fig. 4C (left). For numerical
simulations, we have used the parameters D~0:03, r1~1, r2~1:4and r3~3. The same color coding for the node activity as in Fig. 2
is applied.
(MP4)
Author Contributions
Conceived and designed the experiments: NK ASM. Performed the
experiments: NK. Analyzed the data: NK HK ASM. Contributed
reagents/materials/analysis tools: NK HK ASM. Wrote the paper: NK
HK ASM.
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